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 A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations. 133
 1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism. The EULERi transform (A056156) is also interesting. a(n) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n. - Gus Wiseman, Mar 17 2017 LINKS Aliaksandr Siarhei, Table of n, a(n) for n = 1..102 Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394. Peter J. Cameron, Problems on Permutation Groups, see Problem 3 Index entries for sequences related to binary matrices. FORMULA Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform. a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n. EXAMPLE E.g. a(2) = 3: two ones in same row, two ones in same column, or neither. a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9. There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation: [1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1] [0 1 0] [0 0 1] [1 0] [1 0] ....... [1]. [0 0 1] ....... [0 1] ............. [1] Non-isomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)). - Gus Wiseman, Mar 17 2017 PROG (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)} permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m} K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))} a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023 CROSSREFS Main diagonal of A321609. Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724. Cf. A057149, A057150, A057151, A057152. Cf. A034691, A056156, A089259, A116540, A283877. Sequence in context: A052370 A053304 A053305 * A068590 A327736 A319752 Adjacent sequences: A049308 A049309 A049310 * A049312 A049313 A049314 KEYWORD nonn,nice AUTHOR Peter J. Cameron EXTENSIONS More terms and formula from Vladeta Jovovic, Jul 29 2000 a(19)-a(28) from Max Alekseyev, Jul 22 2009 a(29)-a(102) from Aliaksandr Siarhei, Dec 13 2013 Name edited by Gus Wiseman, Dec 18 2018 STATUS approved

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